Optoelectrical Modeling of Perovskite/Perovskite/Silicon Triple‐Junction Solar Cells: Toward the Practical Efficiency Potential

Perovskite‐based triple‐junction solar cells offer the potential for highly efficient and cost‐effective photovoltaic energy conversion. This article aims to provide a roadmap for the optical properties of perovskite/perovskite/silicon triple‐junction cells. A comprehensive optoelectrical model for the perovskite/perovskite/silicon structure is developed in Sentaurus TCAD. The optical part of the model is validated by measurements of a triple‐junction solar cell. As the electrical characterization is an ongoing process, the electrical properties are assumed to be nonlimiting, which enables us to translate the optical improvement steps into efficiency potentials. A first improvement step lies in adjusting the thicknesses of the perovskite layers to achieve current matching between both perovskite subcells. Using perovskites with bandgaps optimized for planar surfaces, it would be possible to increase the photocurrent density to 13.3 mA cm−2 and the efficiency to 41.9%. It is shown that by implementing a fully textured structure and using the best available materials, a short‐circuit current of 14.1 mA cm−2 and an open‐circuit voltage of 3.48 V with an efficiency of 44.3% are possible assuming idealized electrical properties. This can be regarded as a practical efficiency potential for this kind of triple‐junction technology.


Introduction
Metal halide perovskites hold significant promise as costeffective, scalable absorber materials for the future generation of solar cells.The utilization of perovskite and perovskite/silicon (PS) tandem cells has the potential to decrease the levelized cost of electricity (LCOE) by 10-20% compared to conventional silicon single-junction cells. [1]Moreover, the tunable bandgaps of perovskites make them ideal candidates for constructing multijunction cells with high efficiency.The triple-junction cell architecture with a silicon bottom cell has the detailed balance limit of 49.6%, [2] making the perovskite/perovskite/Silicon (PPS) cell a good candidate for a high-efficiency, low-cost solar cell.Alongside all-perovskite cells, [3][4][5] the construction of PPS cells [5][6][7][8] has already been demonstrated.Their development and processing bring with it additional complexity.To realize the full potential of PPS cells, new perovskite compositions with suitable bandgaps, as well as electron and hole transport layers (ETLs/HTLs), must be used.Furthermore, the layer deposition techniques must be adapted accordingly.
The PPS cell architecture is relatively new, with the first report in 2018 by Werner et al. [7] demonstrating an efficiency of ≈14%.The authors predicted the potential of over 30% for their cell if the suitable low-and high-bandgap perovskite could be realized.Subsequent advancements led to an efficiency of 20% in 2022 [8] and to 22% in 2023. [6]Hörantner, Leijtens et al. [5] predicted a realistic efficiency potential of 39% for a PPS solar cell.However, their work only uses the transfer-matrix-method (TMM) for the optical simulation and a diode model for the electrical simulation.The number of physical phenomena described by that is limited.Also, a detailed loss analysis is not given.However, simulations can assist in the process of optimizing a solar cell.In particular, current matching in multijunction cells is not trivial and yet has great potential for achieving higher efficiencies and can be guided by simulation work.
In this work, we elaborate on a comprehensive optoelectrical simulation model for PPS solar cells, accurately describing the optical properties of state-of-the-art PPS devices.Our optical simulation is validated with measured external quantum efficiency (EQE) and reflectance data from a state-of-the-art PPS cell, [9] ensuring its accuracy in describing the cell's optical behavior.Based on our simulation, we present an efficiency roadmap for optimizing the optical properties of the PPS cell within realistic boundary conditions.The roadmap includes the adaption of the perovskite absorber thicknesses, modifying bandgaps, employing a fully textured cell, and optimizing the thicknesses of the interlayers between the absorbers.We calculate the respective photocurrent of each step and compare it against the theoretical limit.To translate the individual steps into efficiency potentials, an electrical simulation model in Sentaurus TCAD [10] was elaborated for this triple-junction technology for the first time including the ion migration within both perovskite absorbers.As experimental characterization of the device is an ongoing process, this article is restricted on idealized electrical properties, which allow us to yield a practical efficiency potential for PPS solar cells analogously to a previous publication by Er-Raji et al. [11] .Full electrical simulation and experimental validation of the current PPS solar cell remain for future work.
The purpose of the presented roadmap is to give a guideline for the prospective development of PPS solar cells.

Modeling Approach
The optical modeling of the triple-junction cell requires different simulation methods for different regions within the cell.In layers with thicknesses comparable to the wavelength of incoming light, coherent treatment is necessary, which is achieved using the TMM.On the other hand, the silicon layer, with thicknesses above 100 μm, must be treated incoherently, and ray tracing is employed.
[13] In this study, we extend the model to the triple-junction solar cell, which includes an additional perovskite absorber layer with ETLs/HTLs.To account for the different bandgaps in the top and middle perovskite layers, we need to approximate the nk-data, due to the lack of experimental data.We do this by shifting nk-values from a dataset by Manzoor et al. [14] to the desired bandgap on the energy axis (see Appendix for details).However, due to an uncertainty of at least 0.01 eV in the absolute bandgap position, there is a margin of error in the determination of the ideal bandgap.A comprehensive list of optical parameters can be found in Table A1.
In cases where only the properties of the top and middle cells are of interest, and the silicon subcell does not limit the current, a simplified model is employed, treating the silicon as a semiinfinite perfect absorber.This simplification eliminates the need for ray tracing, significantly speeding up computation time and allowing exploration of a broader parameter space.For planar front cases, the simplified model based on TMM Python code [15] can be used, which has been validated to yield the same results to the comprehensive Sentaurus TCAD model.However, when a precise description of the optical properties of the silicon is necessary, the full model, including ray tracing, is used.
To assess the practical efficiency potential of PPS solar cells, we extend our electrical PS tandem model [16] to a triple-junction model and perform electrical simulations.The electrical model calculates a generation profile from the absorption profile calculated by the optical model.It is based on the Poisson and drift-diffusion equations for the electron and holes, as well as for mobile anions and cations within both perovskite absorbers.Sentaurus TCAD allows for sophisticated physical models describing the radiative recombination, Shockley-Read-Hall (SRH), and Auger recombination.At the recombination junctions, band-toband tunneling models are used to describe the interconnection between the subcells.Additionally, series and parallel (shunt) resistances can be implemented by an external circuit model.For detailed information on the electrical model, refer to Table A2.

Experimental Validation
The validation of our optical model is based on experimental data obtained from a monolithic PPS solar cell (in the following sections called the original cell). [9]The original PPS cell's layer stack, as shown in Figure 1a, consists of a silicon heterojunction cell on top of an indium tin oxide (ITO) layer with silver as back contact layer.The silicon features a structured backside with random pyramids, while the front side remains planar.Between the middle and bottom cell, ITO functions as the recombination layer, accompanied by PTAA/PFN as the middle cell's HTL.The middle cell comprises Cs 0.05 (FA 0.90 MA 0.10 ) 0.95 Pb(I 0.95 Br 0.05 ) 3 with a bandgap of 1.57 eV.C 60 serves as the ETL of the middle cell, and SnO x acts as a buffer layer.The top perovskite, consisting of Cs 0.05 (FA 0.55 MA 0.45 ) 0.95 Pb(I 0.55 Br 0.45 ) 3, has a bandgap of 1.84 eV using 2PACz as HTL and C 60 /SnO x as ETL/buffer layer.The top ITO functions as the transparent conductive oxide (TCO) for lateral conduction to the fingers.An antireflection coating comprising MgF 2 is applied on top.
To validate the optical simulation, the simulated data (solid line) is compared to the measured one (dotted lines), as shown in Figure 1b as function of the wavelength.The simulated reflectance is plotted together with the measured reflectance, and the simulated photocurrent is compared with the measured EQE.The colored areas in the graph represent the absorption of the corresponding layers.
The layer thicknesses in the model were initially set to experimental target values.However, the thicknesses of the thin layers and perovskite layers show deviations over the deposited area and may differ from the target thicknesses.We therefore adjusted the simulated thicknesses in a way that agreement with measured EQE and reflectance values is achieved while deviations from the target thicknesses were kept as small as possible.Layer thickness values are given in the appendix (Table A3).Notably, relative deviations remain small, except for very thin layers (below 20 nm).However, the absolute deviation from the experimental target to the fit simulation thickness values remains within 10 nm for thin layers, which is within the magnitude of the inherent error in thickness estimation and lateral inhomogeneities that can be seen in scanning electron microscope (SEM) images.[19] This can be accessed, for example, by spectrometric characterization. [20,21]Since the photocurrent of limiting subcell, which is the middle cell, is known, we scaled the middle EQE in Figure 1b to match the measured value of the short-circuit current of 8.9 mA cm À2 .The other EQEs remain unscaled.The simulated photocurrent of the limiting middle cell is calculated to be 8.7 mA cm À2 .
Regarding reflectance, the simulation aligns well with experimental data except for wavelengths of <350 and ≈900 nm.The deviation for long wavelengths can result from laterally inhomogeneous perovskite thicknesses, visible in SEM images, [9] causing variations of about 10% in thickness.Different perovskite thicknesses can shift the reflection maxima along the wavelength axis.Also, the ray tracing completely neglects wave optics, which could also have an additional impact.At ≈300 nm, deviation can arise from the slightly rough surface of the experimental structure compared to the simulation's assumption of a perfectly planar front side.This deviation has however negligible impact due to the reference AM1.5G spectrum being close to 0 around 300 nm.In the silicon bottom cell, differences of EQE and simulated photocurrent above 900 nm may stem from the difference in reflectance, as the ratio of photons not reflected to absorbed is similar.
Overall, our simulation results exhibit a close alignment with experimental data.Notably, the simulation discloses a significant limitation in the photocurrent of the middle cell.

Optimization of Cell Optics
With the successful validation of the optical model through the agreement between simulated and experimental data in Section 3, we proceed to use the model to optimize the solar cell structure.The optimization of the cell optics focuses on maximizing the photocurrent under technologically viable boundary conditions, namely, the enhancement of the lowest photocurrent among the three subcells, in order to maximize the total current output of the monolithic PPS cell by so-called current matching.
Our approach to this optical optimization involves a sequence of steps (1-4) that progressively build upon each other, following a logical technological realization.These steps, elaborated in more detail within Subsections 4.1-4.4,are as follows: 1) As a first measure, we vary the absorber thicknesses while keeping the perovskite bandgaps constant as modifying layer thicknesses in a certain range is, compared to bandgap variation, relatively straightforward.This step aims to achieve current matching between the top perovskite cells, while the silicon bottom cell yields a nonlimiting high photocurrent density exceeding 16 mA cm À2 .Changing the thickness of the absorber changes the total amount of light absorbed and therefore the photocurrent of each subcell.It is therefore possible to reduce the photocurrent by thinning the absorber and increase the photocurrent by increasing the absorber thickness; 2) To achieve current matching across all three subcells, adjustment of the perovskite's bandgaps is needed, particularly of the middle cell.This changes the absorption profile depending on the wavelength of the incoming light.By lowering the bandgap energy, it is possible to absorb photons at higher wavelengths, which in turn cannot be absorbed by the next absorber.In this way, the photocurrent can be shifted from one absorber to the next by changing the bandgap.Changing the bandgaps results in different optimized thicknesses for the top and middle perovskite layers.While tuning the bandgaps is possible, it is more challenging in practice due to the need to find suitable and stable chemical compositions with bandgaps suitable for top and middle cell in triple-junction structures.We first adjust only the middle cell bandgap (3a) and then both bandgaps (3b); 3) Introducing a textured front side, as opposed to a planar front, serves to mitigate reflection losses.This modification also increases the path length of the incident light, enhancing photogeneration within the desired absorber layers; and 4) Thinner interlayers are assumed which are considered to be technologically feasible while preserving their electrical properties.This leads us to a practical current density potential for the PPS technology.
These sequential steps allow us to assess the practical potential of photocurrent density of PPS triple-junction cells and conduct an analysis of optical losses and strategies to mitigate them.Utilizing these optimized optical properties, we proceed to evaluate the practical efficiency potential of the current device architecture.

Adapting the Absorber Thicknesses
Starting with the original cell (shown in Figure 1), we first vary the absorber thicknesses.
Adapting the perovskite absorber thicknesses is rather simple but has the potential to achieve current matching between the top and middle cell.This way the current can be improved significantly.But first, the silicon absorber is reduced from 250 to 150 μm, following the current industry trend toward thinner wafers.Note that this trend has been extending to even thinner dimensions, with the current world record using an 130 μm absorber. [22]This thickness for the silicon cell was used throughout the rest of this work.Even though we use a thinner silicon layer, the current density of the triple-junction cell is not limited by the silicon cell in our case, consistently exceeding an absorbed photocurrent of 16 mA cm À2 .Therefore, to optimize the photocurrent in this case, a current matching only between the two perovskites is necessary, enabling the utilization of a simplified model (TMM þ treating the silicon as a perfect absorber) to lower calculation times.The optimized thicknesses we get from using the simplified model are then input for the comprehensive Sentaurus simulation.Both models yield the same results for the thin-film layers.
In all optimization steps, we also adapt the antireflection coating layer.Here it should yield minimal reflection for both perovskite absorbers, which is in the range of 350-850 nm.This is achieved at a thickness of 90 nm for the antireflection layer instead of 130 nm for the original cell.
The dependence of the photocurrent of the limiting subcell on the top and middle cell thicknesses is illustrated in Figure 2. Increasing the middle cell thickness leads to higher absorption of photons within the 650-800 nm range (as depicted in Figure 1b).This increased absorption in the middle cell, and consequently reduced photon absorption in the silicon subcell, result in greater absorption for the combined top and middle cell system, leading to a higher potential photocurrent.Another reason to go for a thicker perovskite layer is that deposited perovskite layers feature lateral inhomogeneous thicknesses.If the material is too thin (too thick) at a certain position, the top cell absorbs too few (too many) photons and the respective subcell limits the device current.Figure 2 demonstrates that higher thicknesses provide greater tolerance to absolute variations in subcell thickness.
Nevertheless, we limit the middle cell thickness to 600 nm for two key reasons.First, it is feasible to achieve such thicknesses with different perovskite deposition techniques.Even though, thicker layers have been reported, [23] depending on the deposition method, it becomes more challenging to maintain consistent morphology along with increasing the thicknesses.Second, j SC gains above 600 nm become smaller (see Table A4) and are partly compensated by V OC losses for thicker perovskite layers due to recombination losses that increase for higher perovskite thicknesses above the diffusion length.
Regarding the top cell thickness, a shift from thickness of the original cell (≈270 nm) is necessary.Achieving current matching between the top and middle cells requires a top cell thickness of d top = 167 nm, as emphasized in Figure 2 (considering a 600 nmthick middle cell, as discussed earlier).This adjustment yields a photocurrent of 11.8 mA cm À2 , an improvement of around 2.9 mA cm À2 .The increase in the current density of the current-limiting middle cell originates from the higher tail of the cell's absorption profile within the 450-650 nm region.
Notably, our simulation shows that a mere change in absorber thicknesses cannot yield current matching across all three subcells for the currently used combination of bandgaps.Therefore, the reduction of the middle cell bandgap is inevitable and leads us to the next section.

Adapting Thicknesses and Bandgaps of the Perovskite Absorber
The next optimization step involves the adjustment of perovskite bandgaps.By permitting changes in bandgaps, particularly for the middle cell, current matching across all cells can be achieved.Current matching is made possible due to the shift of photocurrent from the silicon cell, which possesses a high photocurrent density of 16.3 mA cm À2 in the original cell (as depicted in Figure 1b), to the middle cell.In this instance, the full TCAD model is necessary, incorporating ray tracing within the silicon absorber.Since current matching across all cells is reached here, the antireflection coating layer is now optimized to yield minimum reflection across the whole spectrum, which is reached at ≈105 nm.Our strategy is maintaining the middle cell thickness at a fixed value of 600 nm while varying the middle cell's bandgap, given that this parameter directly influences the current in the silicon cell.Specifically, the silicon bottom cell attains the desired one-third of the total generated photocurrent at a middle cell bandgap of bg mid = 1.47 eV.
Once these parameters are set, the thickness and bandgap of the top perovskite must be adjusted.This is achieved through a swift parameter sweep using the simplified TMM model, similar to the previous scenario.While this is an approximation, as reflection minima and maxima shift along the wavelength axis for varying perovskite thicknesses, predominantly affecting the silicon bottom cell's current, deviations in photocurrent were assessed using the full TCAD model and were found to have minimal impact.
Figure 3 provides insight into the resulting overall limiting photocurrent density for varying the top cell's thickness and bandgap.Evidently, for the current selection of a bandgap of bg top = 1.84 eV, mirroring the original device of Figure 1, the necessary thickness for the top cell aligns with d top = 235 nm (point (a) in Figure 3), which is close to the original cell thickness.This already achieves current match between all cells without changing the top cell's bandgap, underlining that the middle cell bandgap is one of the key challenges for current matching of the three subcells.Nonetheless, for similar reasons as for the middle cell, a greater top cell thickness would be advantageous.As there are various parameter sets that yield current matching (gray line in Figure 3), we could also choose a set yielding a higher V OC .With a thickness of d top = 600 nm, the optimal bandgap for the top perovskite to achieve current matching (as illustrated by the blue line in Figure 3) is bg top = 1.98 eV, which is also accompanied by an increase in V OC .This choice results in a generated photocurrent of 13.3 mA cm À2 .
Perovskite bandgaps can be tuned from 1.2 to 3.0 eV. [24]For the middle cell either FAPBI 3 perovskite with 1.47 eV or Sn-containing perovskites could be employed. [4,25,26]For the topcell mixed-halide perovskites as well as all-inorganic perovskites, a bandgap from 1.8 to 2.0 eV has been reported in literature and their application in triple-junction solar cells has been shown. [3,26]The latest PPS world record cell contains a top cell bandgap within that range. [27]The middle cell bandgap of the PPS world record remains higher (at 1.52 eV) compared to our suggestion of 1.47 eV.Thus, a stable low-bandgap perovskite with good optoelectrical properties remains a challenge.

Adding a Front Texture
Until now, our considerations have been restricted to a planar front structure.However, considerable improvements can be achieved by introducing a front texture.This modification minimizes the total reflection and enhances the optical pathlength within the absorbers.Figure 4 illustrates the absorption profile for the optimized cell featuring a front texture.Due to altered reflectance and the increased pathlength, slight adjustments become necessary for the bandgaps and thicknesses.This leads us to follow the same sequential steps as in the previous scenario (4.2): we establish the middle and top absorber thicknesses at the desired 600 nm value, followed by adjusting the perovskite bandgaps to achieve current matching.Consequently, we determine a middle cell bandgap of bg mid = 1.46 eV (À0.01 eV compared to planar case) and a top perovskite bandgap of bg top = 1.97 eV (À0.01 eV compared to planar case).
The incorporation of the texture enhances the cell's photocurrent density to j ph = 13.9 mA cm À2 in all subcells.

Thinner Interlayers
In a final step, we push the limits of material thinning within the cell stack while maintaining our current processing capabilities and retaining their electrical functions.In the frontal layers, ITO is further reduced to 15 nm, while the buffer layer SnO x is scaled down to 10 nm, and C 60 is set to 5 nm.Likewise, the ETL (C 60 ) positioned between the top and middle cell is set to 5 nm.The ITO layer between top and middle cell as well as below the middle cell is set to 10 nm.They can be made thinner than the top ITO, because there is no need for lateral conductivity.These measures effectively reduce the parasitic absorption of the interlayers to a practical minimum, customized for this particular device architecture.As a result of these refinements, our  optimization culminates in an achievable photocurrent potential of j ph = 14.1 mA cm À2 .

Discussion of the Optical Results
The outcomes of the enhancement steps from the previous subsections, from 4.1 to 4.4, are summarized in Figure 5. Tuning the thickness of the absorbers results in a photocurrent of 11.8 mA cm À2 , an enhancement of 2.9 mA cm À2 compared to the original cell.This is a big improvement considering the relatively straightforward implementation of these changes in the cell structure.Refining the perovskite bandgaps leads to a photocurrent of 13.3 mA cm À2 , which is already 86% of photocurrent of the whole AM1.5G spectrum (15.4 mA cm À2 ) and 92% of the photocurrent of the photons in the AM1.5G spectrum with an energy above 1.12 eV (14.5 mA cm À2 ).This is where the greatest potential for progress lies: achieving the optimal bandgap energies in conjunction with the successful processing of sufficiently thick perovskite absorber layers.Consequently, the adoption of a stable perovskite absorber featuring a middle cell bandgap of bg mid = 1.47 eV, with which current matching across all cells can be achieved, is an essential step in the future development of PPS triple cells.The composition of FAPbI3 with a bandgap of around 1.48 eV which is already used for single-junction solar cells [28] is therefore an interesting candidate for the middle cell.Increasing the top bandgap to bg top = 1.98 eV thereby enhances the V OC even further.The realization of a fully textured cell also brings a considerable improvement (0.6 mA cm À2 ), although its implementation presents notable technological challenges in terms of processing.These steps are compared against the radiative limit, which is the detailed balance limit with only taking radiative recombination into account and considering the boundary conditions of a bottom cell bandgap of 1.12 eV. [2]Notably, the desired perovskite bandgaps for a practical device turn out to be around 30-40 meV lower than estimated by the radiative limit.This indicates that the parasitic absorption within the front side layers and the imperfect light trapping of the perovskite absorbers (which both would require much lower perovskite bandgaps for current matching) are almost counterbalanced by the imperfect light trapping of the silicon absorber and parasitic absorption losses within the rear side layers (which on their own would require much higher perovskite bandgaps for current matching).All these effects are neglected in the calculation of the radiative limit.
Furthermore, stability and degradation are important issues when considering perovskites.However, as long as the degradation mechanisms do not shorten the j SC , the optimization presented here should remain valid.However, it should be emphasized that if we want to produce this type of cell on a large scale, stability issues will need to be resolved for all materials with the respective bandgaps proposed in this article.

Electrical Simulation of PPS Cells: Toward the Practical Efficiency Potential
In this section, the electrical TCAD model for PPS solar cells developed within the scope of this work is now used to evaluate the efficiency potential of the optical roadmap steps presented in the previous section.Before we show the individual roadmap points from Section 4, we add an additional point where the electrical simulation is performed at the original cell's optical parameter, but with idealized electrics to separate the gains from this idealization step from the gains of the individual roadmap points.We thereby neglect SRH and surface recombination, series resistances are minimal, no shunts are assumed, the mobile ion density is assumed negligibly small to suppress hysteresis effects, and the ETLs/HTLs feature ideal energy band alignment for efficient extraction of majority charge carriers and suppressing recombination.The black star on the right is the radiative limit value, that is, one-third of the current density of the AM1.5G spectrum of photons with energy higher than 1.12 eV.The photocurrent in the table below is the one of the entire cell and given by the lowest value of the individual subcells.The major improvements are made by adjusting the thicknesses of the perovskites in order to reach current match between them and adjusting the bandgaps close to the ones predicted by the theoretical limit. [2]eginning with this idealization, there is a notable enhancement in V OC from 2.87 V (original cell) to 3.48 V. Additionally, the fill factor increases to 94.5% and thus, the efficiency rises from 20.1% to 28.5%.The high fill factor is due to the strong current mismatch present between the subcells.This shows the overall potential when all electrical loss channels could be mitigated.In a former publication, we analogously showcased the impact of mitigating each individual loss channel step by step for a PS tandem solar cell based on a variety of measurements with different characterization methods. [11]However, for the PPS cell, this experimental work and establishing equivalent characterization methods is ongoing; therefore, due to the lack of experimental data, we cannot yet assign values to the individual contributions.In Figure 6, a band diagram at the maximum power point of the simulation of the cell in the practical potential scenario with idealized electrics is shown.It can be seen that the quasi-Fermi-level splitting of each subcell adds up to the total V mpp of the cell, indicating the idealized electrical properties of the cell.The cell is shown at the maximum power point after light soaking at open-circuit conditions until quasi steady state.The anions and cations (with initially homogenously distributed ion density N ion = 1 Â 10 16 cm À3 ) gather at the respective contact after being initialized uniformly (right axis in Figure 7).
The J-V parameters of all cell scenarios resulting from the electrical simulation are shown in Figure 7, translating the optical roadmap of Figure 5 into an estimation for the J-V parameters and efficiency: 1) Optimizing perovskite thicknesses (step 4.1 in Figure 7) yield a significant enhancement in short-circuit current density from 8.7 to 11.8 mA cm À2 in line with the optical roadmap.Due to the lower top cell thickness, the V OC slightly increases by 20 mV.The FF drops by 2.5% abs due to current matching of both perovskite absorbers.This is a well-known phenomenon in multijunction solar cells when a current mismatching is dissolved. [21]This adaptation of perovskite absorber thickness translates to a gain of 9.3% abs in efficiency, elevating it to 37.8%; 2) An optimization of perovskite bandgaps (step 4.2 in Figure 7) yields another big improvement due to the increased current density from 11.8 to 13.3 mA cm À2 .As current matching is now achieved between all three subcells, the current density increases while fill factor decreases by another 1.7% abs to 90.1%.The V OC drops from 3500 to 3380 mV when the middle cell bandgap is lowered by ≈100 meV (step 4.2a).But when also adjusting the top bandgap by increasing it around 140 meV, V oc increases to 3490 mV (i.e., almost its initial value of step 4.1), leading to an efficiency of 41.9%. Figure 7 showcases quite nicely that the main motivation for the adaption of the middle cell bandgap is the current matching of the whole device, whereas the main motivation for the adaption of the top cell bandgap is to further increase the V oc potential; 3) Subsequently, the incorporation of a front texture (step 4.3 in Figure 7) enhances the j SC , accompanied by a slight reduction in V OC due to the lowered ideal bandgaps of the perovskites; and 4) Finally,  considering the thinnest interlayer thicknesses (4.4) possible, parasitic absorption is minimized, leading to a practical potential of current density of 14.1 mA cm À2 .For this scenario, an efficiency of 44.3% is proposed, reaching 89% of the theoretical (i.e., radiative) limit.
The practical potential of the PPS triple cell is also compared with the single silicon cell and PS tandem architectures (Figure 8).As the perovskite-based triple-junction solar cell technology is still at an early stage of development, the efficiency of the current record triple cell of 22.2% [6] is below the record of the silicon single junction of 26.8% [22] cell as well as the current record PS tandem cell of 33.2%. [29]However, as thermalization losses are lower in multijunction cells, the practical potential of the triple-junction cell is the highest.An efficiency of 44.3% represents a 51% rel increase over the potential of the silicon single cell with 29.4% [30] and 11% rel improvement over the PS tandem cell with 39.3%. [11]

Summary and Conclusion
A comprehensive optoelectrical model was developed in Sentaurus TCAD for PPS triple-junction solar cells and experimentally validated by means of measured data of a state-ofthe-art experimental PSS cell which was built in-house.Within this work, we elaborated on an optical roadmap to outline a path for improving the short-circuit current density of PPS to its practical limit.By means of the electrical model, we were able to translate the optical roadmap into efficiency potentials.As subcell characterization and electrical loss analysis of triple-junction solar cells are work in progress, the efficiency roadmap was elaborated based on an idealized electrical scenario to showcase the practical efficiency potential.In a first step, we showed that immediate progress could be made by adjusting only the thicknesses of the perovskite absorbers to yield current matching between both perovskite cells.This would increase the current density from 8.7 to 11.8 mA cm À2 and the efficiency from 28.5% to 37.8% for the current bandgaps.Furthermore, we showed in a second step that a reduction of the middle cell bandgap down to 1.47 eV (assuming 600 nm thickness) is inevitable to achieve current matching between all three subcells.On the other hand, we showed that the range of the top-cell bandgap could be chosen between 1.8 and 2.0 eV, depending on the top cell thickness varying between 200 and 800 nm, respectively, for which each a bandgap/thickness tuple exists to achieve global current matching.Nevertheless, we raised awareness for choosing thicker perovskite layers with higher bandgap to unleash the full V oc potential of the top cell.
We furthermore calculated j sc of 13.3 mA cm À2 for planar front devices (featuring a textured silicon hetero junction rear side), yielding an efficiency potential of 41.9%.Texturing enhances j sc from 0.6 to 13.9 mA cm À2 which translates to 43.5% in efficiency, but also introduces new complexities in processing.In a final step, by minimizing the parasitic absorption to a practical minimum (with still electrically and technologically feasible layer thicknesses), our assessment of the practical efficiency potential reveals an efficiency of 44.3%, achieved through j sc of 14.1 mA cm À2 , V OC of 3480 mV, and a fill factor of 90.1%.According to our simulation, this can be achieved using a top cell with a 1.98 eV bandgap and a middle cell with a 1.46 eV bandgap.This efficiency potential for the PPS triple junction lies 4.8% abs above the analogously calculated efficiency potential of 39.5% for a perovskite/silicon tandem solar cell as published previously by Er-Raji and Messmer et al. and is the key driver for triple-junction technology as compared to tandem devices.) PPS cell record is below the records of silicon and PS tandem cells.However, the practical potential and the theoretical limit of the PPS is higher due to lower thermalization losses.Record values from ref. [22] (silicon), [29] (PS tandem), [27] (PPS), [26] (all perovskite); practical potential values from ref. [30] (silicon), [11] (PS tandem), and this work (PPS); theoretical limit from refs.[2,31].Ray tracing In all thick layers, i.e., air and silicon, see [13] TMM In all thin-film layers including both perovskite absorbers

Phong
To account for the rough morphology of the perovskite top cell improving the light trapping properties of the silicon absorber, as used in ref. [13]  there is an uncertainty of at least 0.01 eV included in the data.We then shift the nk-values on the energy axis according to Where E is the photon energy in the reference dataset, bg ref ¼ 1.66 eV the reference bandgap, bg Ã the new (desired) bandgap, and E Ã the energy value of the new dataset.

Figure 1 .
Figure 1.a) Layer stack of the PPS triple-junction solar cell as investigated in this article.b) Simulated absorption and reflectance against the measured EQE and reflectance.Simulated values are represented by solid lines, while the dotted lines represent the measured ones.FCA is the free carrier absorption within the silicon.The colored areas represent the integrated absorption, responsible for the photocurrent of the corresponding layers.

Figure 2 .
Figure 2. Enhancing the photocurrent of the limiting subcell by variation of the thicknesses of the top and middle perovskite subcells.The bandgaps are thereby fixed at 1.57 and 1.84 eV for the top and middle cell, respectively.The photocurrent of the limiting cell is plotted for a range of thicknesses of the middle perovskite d mid and the top perovskite absorber d top .The gray line indicates the points where the photocurrents of top and middle cell are equal and current matching is achieved between them.The value of the photocurrent is indicated by the colors.'Original cell' is the set of parameters of the original cell and 'adapted thicknesses' the values elaborated in this section of 167 and 600 nm for top and middle cell, respectively.The middle cell thickness is set to 600 nm to keep it technologically feasible (see text).

Figure 3 .
Figure 3. Enhancing the photocurrent of the limiting cell by adjusting thickness d top and bandgap bg top of the top cell.The parameters of the middle cell are fixed to a thickness of d mid = 600 nm and a bandgap of bg mid = 1.47 eV to keep the silicon current at one-third of the total generated current.The gray line indicates the points where the photocurrents of top and middle cell are equal and current matching is achieved between them."Before" is the set of top cell parameters of Section 4.1, whereas a) is only a top cell thickness adjustment and b) is both a thickness and bandgap adjustment of the top cell.The top cell thickness limit is set to 600 nm to keep it technologically feasible (see text in Section 4.1).

Figure 4 .
Figure 4.The absorption profile of the optimized cell featuring textured front and rear sides with adjusted bandgaps and thicknesses of both perovskites.The front texture minimizes the reflectance of the device strongly.A photocurrent of 13.9 mA cm À2 is reached.

Figure 5 .
Figure 5. Generated photocurrent density of the top, middle, and bottom cell for the different scenarios from subsections 4.1-4.4.The black cross on the left shows the measured value of the original cell.The black star on the right is the radiative limit value, that is, one-third of the current density of the AM1.5G spectrum of photons with energy higher than 1.12 eV.The photocurrent in the table below is the one of the entire cell and given by the lowest value of the individual subcells.The major improvements are made by adjusting the thicknesses of the perovskites in order to reach current match between them and adjusting the bandgaps close to the ones predicted by the theoretical limit.[2]

Figure 6 .
Figure 6.Band diagram of the device of the practical potential scenario at the maximum power point (left axis).The depth is given relative to the silicon absorber and the potential relative to the left contact.The mobile ion densities of anions (green) and cations (blue) are shown for both top and middle perovskite absorbers (referring to right axis).

Figure 7 .
Figure 7. Electrical parameters (short-circuit current density j sc , open-circuit voltage V OC , the fill factor FF and the power conversion efficiency η) of the different scenarios of the optical roadmap.The first values of "Original cell" correspond to measurements, while the rest are simulated values.

Table A3 .
Material thicknesses for experimental validation.The experimental target values are compared with the adjusted thicknesses used in the simulation.The errors given for the perovskites are statistical errors given by the lateral inhomogeneities of the respective layers.

Table A4 .
j ph gain for values of d mid > 600 nm.